10 SES 01 E, Learning from Practice
Teachers’ responsiveness to their students’ thoughts and actions can be considered as one of the indicators of high-quality mathematics instruction. Teachers’ proper intervention to their students’ mathematical understanding can only be ensured by their awareness of what their students do and say through instruction. In this context, teachers’ ability of detecting a pedagogically rich classroom opportunity such as, students’ answers of a specific problem, problem solution steps, or explanation of their reasoning, and teachers’ ability of using this opportunity to enhance students’ understanding is defined as teachers’ noticing skills (van Es & Sherin, 2002). Teachers’ noticing skills has become an important part of teacher competence, especially in the last 15 years, and has begun to make room for itself as a separate lane in educational research (Jacobs, Lamb & Philipp, 2010; Mason, 2011; Stockero, Rupnow, & Pascoe, 2017).
Jacobs and colleagues (2010) tried to narrow the scope of noticing skills to a certain degree and focus on noticing specific classroom moments such as students reveal their mathematical thinking. They defined teachers’ ability to notice students' mathematical thinking in three stages: 1) attending students' strategies, 2) interpreting students' insights and understanding, and 3) deciding how to respond based on their interpretations. Firstly, the teacher must pay attention to the verbal and written strategies that students use so as to reveal their mathematical thinking. Then, by passing through his/her own knowledge and experience, s/he must interpret the strategy of the students. And eventually, based on his/her interpretation, s/he should give a proper response in order to improve students’ mathematical thinking.
As a contribution to Jacobs and his colleagues’ framework, one of the questions that needs to be answered in detail is that what are the classroom moments that should be noticed. The framework of Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (MOSTs) can be an efficient answer for this question (Leatham, Peterson, Stockero, & van Zoest, 2015). According to Leatham et al. (2015), the moments that a student present their mathematical thinking which have chance to promote students’ mathematical understanding if it was discussed provides an important pedagogical opportunity. They call these moments as MOSTs. In order to be a MOST case, a classroom moment should i) be based on students’ mathematical thinking, ii) be mathematically important, iii) uncover a pedagogical opportunity. Firstly, it should be possible to make evidence-based interpretation about students' mathematical thinking and this students’ thinking should be related with a mathematics concept. Secondly, the mathematical concept that students thinking highlight must have an important place among mathematics learning objectives of the relevant class level. Lastly, there should be a cognitive need and enough time to discuss this mathematics concepts. If these three basic criteria met then there is a "mathematical opportunity" (Leatham et al., 2015).
In the light of the importance of noticing students’ mathematical thinking and pedagogical opportunities that can be built on these thinking, the aim of this study is to identify MOST cases that emerged in 2 tasks for teaching integers and investigate the possible reasons that make teacher candidates to notice or unable to notice these opportunities. In this proposal we tried to answer followings:
- What are the MOST cases that emerged in instructional tasks designed for teaching integers?
- Are pre-service teachers enable to notice these MOST cases?
- What are the possible reasons that hinder pre-service teachers from responding these opportunities productively?
This research is a case study aiming to investigate both MOST instances occurred in a real classroom environment and identify pre-service teachers’ in-the-moment noticing abilities. It was conducted under a faculty-school collaboration between a large university in Istanbul and a local middle school in the neighborhood. The aim of the collaboration is to support students’ mathematical thinking and understanding as well as pre-service teachers’ professional knowledge and skills. We offered an elective course for pre-service teachers in the university such that we discussed student-centered teaching approaches, task design and implementation and scaffolding practices in this course. As a part of course requirements, pre-service teachers implemented different tasks that was developed by research team in a 7th grade mathematics classroom through 2 weeks (4 lesson hours). After each implementation, there was a reflection session in which each pre-service teacher commented on students’ performances. 8 pre-service teachers and 32 7th grade students participated in the study. As the research team, we made a group of four students and assigned a pre-service teacher for each group for implementation of tasks. The pre-service teachers worked with the same group of students throughout the study. We followed a 3-step process for task implementation: 1) students worked on the given tasks individually, 2) they discussed their solutions within their groups, and 3) pre-service teacher joined in their discussion and asked for their solutions and reasoning. As students were working individually or making group discussions, pre-service teachers observed them and took some notes about their performances. We expected that pre-service teachers would respond to the MOST instances that they observed during individual work and group discussion. The data was collected through videos, reflection reports, assignments and students’ worksheets. We videotaped all pre-service teacher-students interactions and oral reflection sessions. Pre-service teachers wrote a reflection report for each task implementation. We collected students’ worksheets and extra sheets at the end of the implementations. We analyzed each pre-service teacher’s implementation videos to determine MOST instances. Then we attempted to analyze whether the pre-service teacher attended to that MOST instance and how s/he responded to it. We used Leatham and his colleagues’ (Leatham et al., 2015) framework to identify MOST instances in the videos and tried to describe how pre-service teachers acted and interacted with students when they recognized the MOST instance in the line of Jacobs and his colleagues’ (Jacobs et al., 2010) definition of noticing.
The first MOST case was emerged when some of the students in groups wrote -2+3=5 and -2+2=4 to describe their movement in first task (In the first task, students were expected to write mathematical expressions of their movement in school’s stairs). Although pre-service teachers noticed these mistakes, they can’t benefit from this pedagogical opportunity. The reactions of pre-service teachers were remained restricted to either indicating the correct answer or warning students to be more careful. In addition, while reflecting on students’ performances in reflection sessions, preservice teachers mainly complained of students’ carelessness or lack of attention. These complains of carelessness can be an indication of why they responded this most cases unproductively. Because when the problem is asserted as carelessness but not deeper misunderstandings about integers, the proposed solution will be ‘to be more careful’ but not digging students’ misconceptions. The other MOST case was emerged as finding the midpoint of -130 and -270. The students calculated the midpoint as -130/2=-65, -270/2=-135, (-65) + (-135) = -195. As a reaction, the pre-service teacher communicated her own solution strategy, “lets try to find the distance between -130 and -270, it is 140. What is the middle of 140, it is 70. When subtract 70 from -130, it is -200”. In this MOST case, since pre-service teacher stuck into her solution, she couldn’t benefit from her student’s novel strategy. It can be claimed that pre-service teachers’ obsession of ‘correct answer’ can become an obstacle in responding productively to this pedagogical opportunity.
Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202. Leatham, K. R., Peterson, B. E., Stockero, S. L., & van Zoest, L. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education, 46(1), 88-124. Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). New York: Routledge. Stockero, S. L., Rupnow, R. L., & Pascoe, A. E. (2017). Learning to notice important student mathematical thinking in complex classroom interactions. Teaching and Teacher Education, 63, 384-395. Van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571–596.
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